This paper investigates the potential of an axial velocity electron beam, to generate slow waves and to become azimuthally bunched. The electron beam is consider- ed to be initially unmodulated, monoenergetic, with no initial transverse velocity of each constituent electron, infinitesimally thin and at low current. The electrons are acted upon by a guided mode, under conditions of cyclotron resonance. The nonlinear dynamics of the electron beam is described by means of the equations of motion, which have been found to be integrable, since they exhibit two integrals of motion. The first invariant gives the conditions for the beam voltage to decrease below its ini- tial value. For initially vanishing transverse electron momentum (u10=0), this occurs when the participating waves have phase velocities less than the speed of light in vacuum (refractive index n=kc/w>1). The necessary conditions correspond to the anoma- lous Doppler shift. The appropriately normalized expression for the frequency mis- match A=(Q/Yo-io+kvo )/w and the radiation field amplitudes have been obtained and it was found that generation of substantial transverse momentum is possible, especially if A>0. Upon performing an appropriate optimization to maximize u1, expressions for the expected output transverse momentum, as well as for the associated efficiency and the required normalized frequency mismatch, have been obtained.